Theorems · Theorem · commutative algebra
IsBezout.iff_span_pair_isPrincipal
∀ {R : Type u} [inst : CommRing R], IsBezout R ↔ ∀ (x y : R), Submodule.IsPrincipal (Ideal.span {x, y})- Defined in
- Mathlib.RingTheory.Bezout
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Submoduleproof · cited by 7,192
- Submodule.spanproof · cited by 1,504
- Ideal.spanstatement and proof · cited by 948
- Submodule.FGproof · cited by 230
- Submodule.IsPrincipalstatement and proof · cited by 129
- IsBezoutstatement and proof · cited by 23
- Submodule.IsPrincipal.casesOnproof · cited by 15
- Submodule.span_insertproof · cited by 6
- Submodule.fg_inductionproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- Function.Surjective.isBezoutproof · cited by 0